Does this require a given hardware platform, or for example, could I run this model with a cheap mini robot with a single camera (i.e. good enough without depth), tracked treads, and an arm?

When the page says "embodiment-free (UMI)" it means they've used a standardised gripper and camera arrangement [1], while aiming to be agnostic to the type of robot arm that gripper is attached to.

[1] https://umi-gripper.github.io/

I’d imagine the training is hardware specific